Correlated Fermions with Spin-Orbit Coupling and Competing Orders in Solids

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Correlated electrons in solids exhibit a huge variety of fascinating phenomena ranging from magnetism and metal-insulator transitions to high-temperature superconductivity. The project aims at systematically investigating the phase diagram of interacting fermion systems in the context of condensed matter physics. Dirac materials, such as graphene, and a growing number of novel two-dimensional systems exhibit a huge variety of possible ordered states in the presence sufficiently strong interactions. We shall explore effective models to analyze the competition of these order parameters and quantitatively determine their critical behaviour close to the intersection of various phase transition lines. In many novel materials and in topological states of matter, a finite spin-orbit coupling plays a crucial role. At the same time electron-electron interactions can be sizable and the interplay between correlation effects and spin-orbit coupling is only beginning to be understood. We will develop a versatile approach to identify emerging many-body instabilities aiming at an unbiased theoretical description of the low-energy properties of (two-dimensional) materials with sizable spin-orbit coupling. The project will be studied with modern formulations of the Renormalization Group.